Viable control of an epidemiological SIR model

We consider vaccination control of the spread of an epidemy in a classical SIR model. Our approach aims at controlling the number infected at the peak. It differs from the widespread stationary vaccination control strategies, based upon having control reproductive number stricly less than one to ensure convergence, and also from cost minimization optimal control ones. Indeed, instead of aiming at an equilibrium or optimizing, we look for policies able to maintain the number of infected individuals below a threshold for all times. Thus doing, we focus both on transitories and on asymptotics, in a robust way. We provide a formulation of an epidemy management as a dynamic control under constraint problem, for which the constraint to maintain the number of infected individuals below a threshold for all times has to be achieved by a time-dependent vaccination strategy. The so-called viability kernel is the set of initial states for which such a vaccination policy exists. We give an expression of the viability kernel, and characterize viable policies. We exhibit policies that are both viable and asymptotic, in that they both control the maximum number infected at the peak and asymptotically drive the number of infected to zero.